1 , . . . , λ C n with rational coeﬃcients of H 1 , 1 prim ( X, Q ) , that is, there are n ∶ = h 1 , 1 prim ( X, Q ) algebraic curves C 1 , . . . , C n in X such that under the Poincar´e duality the class in homology [ C i ] goes to λ C i , [ C i ] ↦ λ C i . Recall that the Cox ring of P k + 2 is contained in the Cox ring of P 2 k + 1 Σ ,X without considering the grading. Considering the grading we have that if α ∈ Cl ( P k + 2 Σ ) then ( α, 0 ) ∈ Cl ( P 2 k + 1 Σ ,X ) . So the polynomials deﬁning C i ⊂ P k + 2 Σ can be interpreted in P 2 k + 1 X, Σ but with diﬀerent degree. Moreover, by Remark 4.1 each